Definition
Algebraically Complete Field
A field
is algebraically complete (or algebraically closed) if every non-constant polynomial with coefficients in has at least one root in . Formally: for every
with , Equivalently, every
splits into linear factors over :
Examples
is algebraically complete By the Fundamental Theorem of Algebra, every non-constant polynomial with complex coefficients has a complex root. Hence
is algebraically complete.
is not algebraically complete The polynomial
has no real root, since for all .